A direct numerical simulation of a turbulent channel flow is performed. The unsteady Navier-Stokes equations are solved numerically at a Reynolds number of 3300, based on the mean centreline velocity and channel half-width, with about 4 × 106 grid points (192 × 129 × 160 in x, y, z). All essential turbulence scales are resolved on the computational grid and no subgrid model is used. A large number of turbulence statistics are computed and compared with the existing experimental data at comparable Reynolds numbers. Agreements as well as discrepancies are discussed in detail. Particular attention is given to the behaviour of turbulence correlations near the wall. In addition, a number of statistical correlations which are complementary to the existing experimental data are reported for the first time.

Numerical simulations of fully developed turbulent channel flow at three Reynolds numbers up to Reτ=590 are reported. It is noted that the higher Reynolds number simulations exhibit fewer low Reynolds number effects than previous simulations at Reτ=180. A comprehensive set of statistics gathered from the simulations is available on the web at http://www.tam.uiuc.edu/Faculty/Moser/channel.

A direct numerical simulation of incompressible channel flow at Re τ = 5186 has been performed, and the flow exhibits a number of the characteristics of high Reynolds number wall-bounded turbulent flows. For example, a region where the mean velocity has a logarithmic variation is observed, with von Kármán constant κ = 0.384 ± 0.004. There is also a logarithmic dependence of the variance of the spanwise velocity component, though not the streamwise component. A distinct separation of scales exists between the large outer-layer structures and small inner-layer structures. At intermediate distances from the wall, the one-dimensional spectrum of the streamwise velocity fluctuation in both the streamwise and spanwise directions exhibits k −1 dependence over a short range in k. Further, consistent with previous experimental observations, when these spectra are multiplied by k (premultiplied spectra), they have a bi-modal structure with local peaks located at wavenumbers on either side of the k −1 range.

One-point statistics are presented for new direct simulations of the zero-pressuregradient turbulent boundary layer in the range Re θ = 2780-6680, matching channels and pipes at δ + ≈ 1000-2000. For tripped boundary layers, it is found that the eddyturnover length is a better criterion than the Reynolds number for the recovery of the largest flow scales after an artificial inflow. Beyond that limit, the integral parameters, mean velocities, Reynolds stresses, and pressure fluctuations of the new simulations agree very well with the available numerical and experimental data, but show clear differences with internal flows when expressed in wall units at the same wall distance and Reynolds number. Those differences are largest in the outer layer, independent of the Reynolds number, and apply to the three velocity components. The logarithmic increase with the Reynolds number of the maximum of the streamwise velocity and pressure fluctuations is confirmed to apply to experimental and numerical internal and external flows. The new simulations also extend to a wider range of Reynolds numbers, and to more than a decade in wall distance, the evidence for logarithmic intensity profiles of the spanwise velocity and of the pressure intensities. Streamwise velocity fluctuations appear to require higher Reynolds numbers to develop a clear logarithmic profile, but it is argued that the comparison of the available near-wall data with fluctuation profiles experimentally obtained by other groups at higher Reynolds numbers can only be explained by assuming the existence of a mesolayer for the fluctuations. The statistics of the new simulation are available in our website.

The spectra and correlations of the velocity fluctuations in turbulent channels, especially above the buffer layer, are analysed using new direct numerical simulations with friction Reynolds numbers up to Re$_{\tau}\,{=}\,1900$. It is found, and explained, that their scaling is anomalous in several respects, including a square-root behaviour of their width with respect to their length, and a velocity scaling of the largest modes with the centreline velocity $U_c$. It is shown that this implies a logarithmic correction to the $k^{-1}$ energy spectrum, and that it leads to a scaling of the total fluctuation intensities away from the wall which agrees well with the mixed scaling of de Graaff & Eaton (2000) at intermediate Reynolds numbers, but which tends to a pure scaling with $U_c$ at very large ones.

A study of compressible supersonic turbulent flow in a plane channel with isothermal walls has been performed using direct numerical simulation. Mach numbers, based on the bulk velocity and sound speed at the walls, of 1.5 and 3 are considered; Reynolds numbers, defined in terms of the centreline velocity and channel half-width, are of the order of 3000. Because of the relatively low Reynolds number, all of the relevant scales of motion can be captured, and no subgrid-scale or turbulence model is needed. The isothermal boundary conditions give rise to a flow that is strongly influenced by wall-normal gradients of mean density and temperature. These gradients are found to cause an enhanced streamwise coherence of the near-wall streaks, but not to seriously invalidate Morkovin's hypothesis : the magnitude of fluctuations of total temperature and especially pressure are much less than their mean values, and consequently the dominant compressibility effect is that due to mean property variations. The Van Driest transformation is found to be very successful at both Mach numbers, and when properly scaled, statistics are found to agree well with data from incompressible channel flow results.

Three direct numerical simulations of incompressible turbulent plane mixing layers have been performed. All the simulations were initialized with the same two velocity fields obtained from a direct numerical simulation of a turbulent boundary layer with a momentum thickness Reynolds number of 300 computed by Spalart [J. Fluid Mech. 187, 61 (1988)]. In addition to a baseline case with no additional disturbances, two simulations were begun with two-dimensional disturbances of varying strength in addition to the boundary layer turbulence. After a development stage, the baseline case and the case with weaker additional two-dimensional disturbances evolve self-similarly, reaching visual thickness Reynolds numbers of up to 20 000. This self-similar period is characterized by a lack of large-scale organized pairings, a lack of streamwise vortices in the ‘‘braid’’ regions, and scalar mixing that is characterized by ‘‘marching’’ probability density functions (PDFs). The case begun with strong additional two-dimensional disturbances only becomes approximately self-similar, but exhibits sustained organized large-scale pairings, clearly defined braid regions with streamwise vortices that span them, and scalar PDFs that are ‘‘nonmarching.’’ It is also characterized by much more intense vertical velocity fluctuations than the other two cases. The statistics and structures in several experiments involving turbulent mixing layers are in better agreement with those of the simulations that do not exhibit organized pairings. .

No abstract